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# Re: st: Interpretation of interaction term in log linear (non linear) model

 From Suryadipta Roy To statalist@hsphsun2.harvard.edu Subject Re: st: Interpretation of interaction term in log linear (non linear) model Date Mon, 10 Jun 2013 09:37:53 -0400

```Dear David,
Thank you very much for the wonderful suggestions! I would make sure
that the writeup reflects them! In fact, I have used -margins- to
reflect the contribution of x1 in the two groups as well. A related
question was if the coefficient of the interaction term in the
log-linear model can be interpreted as a log-odds ratio, and that an
important property of odds ratios is that it is constant, i.e. does
not matter what values the other independent variables take on. I
believe that your suggestions (and my query here) is relevant for
Poisson/NB models as well.

Sincerely,

On Sun, Jun 9, 2013 at 1:58 PM, David Hoaglin <dchoaglin@gmail.com> wrote:
>
> It may be helpful to focus, initially, on the fitted model in the log scale.
>
> The definition of the coefficient of "dummy" includes the list of
> other predictors in the model (constant, x1, and dummy*x1).  Also,
> when you interpret the coefficient of "dummy", you should mention that
> it summarizes the effect of "dummy" on log(Trade) after adjusting for
> simultaneous linear change in x1 and dummy*x1.  If the interpretation
> of the coefficient of "dummy" does not mention those adjustments, it
> gives the impression that the coefficient summarizes the change in
> log(Trade) corresponding to an increase of 1 unit (i.e., 1 SD) in x1
> when the other predictors are held constant.  In your model that
> oversimplified interpretation is misleading, because one cannot change
> "dummy" and hold dummy*x1 constant.  More generally, the "held
> constant" interpretation does not reflect the way multiple regression
> works.
>
> The presence of the interaction term implies that the model makes
> separate adjustments for the contribution of x1 in the two groups
> defined by "dummy".  It also implies (as you mentioned) that the
> effect of "dummy" depends on the value of x1.  It is easiest to
> calculate that effect when x1 = 0.  That may be an appropriate
> starting point, but you should also show the mean of x1 when "dummy" =
> 0 and the mean of x1 when "dummy" = 1 (and look at the relation
> between the ranges of x1 in the two groups).  Centering the variable
> underlying x1 is likely to be a good idea, but the case for dividing
> by its standard deviation is less clear.
>
> This discussion should clarify the interpretation and provide a basis
> for translating it to the original scale of the data.  It applies also
> if you use quasi-likelihood, conveniently available in the glm/poisson
> framework.  If you want to work in terms of elasticities, please check
> that any derivatives involved do not (inappropriately) assume that the
> other predictors can be held constant.
>
> David Hoaglin
>
> On Sat, Jun 8, 2013 at 12:12 PM, Suryadipta Roy <sroy2138@gmail.com> wrote:
>> Dear Statalisters,
>>
>> I was wondering if some one would be kind enough to clarify if I am on
>> the right track in clarifying the coefficient of the interaction term
>> when the dependent variable is in logarithm. The estimated model is of
>> the form: log(Trade) = constant + 0.15dummy - 0.15x1 + 0.12dummy*x1,
>> where dummy is (0,1) categorical variable, x1 is a continuous variable
>> (standardized 0 - 1), and dummy*x1 is the interaction term. The result
>> has been obtained from a fixed effects panel regression using -areg-
>> with robust standard error option, and all the variables are
>> statistically significant. Based on readings of Maarten's Stata tip
>> 87: Interpretation of interactions in non-linear model, several
>> Statalist postings, and the following link
>> http://www.stanford.edu/~mrosenfe/soc_388_notes/soc_388_2002/Interpreting%20the%20coefficients%20of%20loglinear%20models.pdf
>> , I wanted to make sure if any of the following interpretation of the
>> above result is correct:
>>
>> 1. The coefficient of "dummy" indicates that this category (dummy
>> variable = 1) has 16% (= exp(0.15) - 1) more of "Trade" compared to
>> the base category (dummy variable = 0).
>> 2. The effect of being in this category on "Trade" increases when the
>> value of x1 increases. For every standard deviation increase in x1,
>> the effect of "dummy" increases by about 13% (exp(0.12) - 1), OR there
>> is a statistically significant 13% increase in "Trade" to countries
>> having more of x1 relative to countries that have one standard
>> deviation lower value of x1, OR the effect of being in the "dummy = 1"
>> category in a country with one standard deviation more of x1 than
>> average is exp(0.12)*exp(0.15) = 1.31, which means that "dummy=1"
>>
>> Following suggestions elsewhere in the Statalist, I have pursued other
>> non-linear estimation strategies (and have asked questions to that
>> effect earlier), but there is a tradition in this literature to use
>> log-linear models. Any suggestion is greatly appreciated.
>>
>> Sincerely,
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